Global optimization method for design problems

Engineering Review, Vol. 36, Issue 2, 149-155, 2016.

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GLOBAL OPTIMIZATION METHOD FOR DESIGN PROBLEMS Abdelkhalak El Hami1 – Seifedine Kadry2* 1 2

*

INSA of Rouen, Normandy University, France Engineering School, American University of the Middle East, Kuwait.

ARTICLE INFO

Abstract:

Article history: Received: 13.2.2015. Received in revised form: 24.3.2015. Accepted: 27.3.2015. Keywords: Global optimization Simulated annealing (SA) Triplate Structural optimization Simultaneous perturbation stochastic approximation (SPSA)

In structural design optimization method, numerical techniques are increasingly used. In typical structural optimization problems there may be many locally minimum configurations. For that reason, the application of a global method, which may escape from the locally minimum points, remains essential. In this paper, a new hybrid simulated annealing algorithm for global optimization with constraints is proposed. We have developed a new algorithm called Adaptive Simulated Annealing Penalty Simultaneous Perturbation Stochastic Approximation algorithm (ASAPSPSA) that uses Adaptive Simulated Annealing algorithm (ASA); ASA is a series of modifications done to the traditional simulated annealing algorithm that gives the global solution of an objective function. In addition, the stochastic method Simultaneous Perturbation Stochastic Approximation (SPSA) for solving unconstrained optimization problems is used to refine the solution. We also propose Penalty SPSA (PSPSA) for solving constrained optimization problems. The constraints are handled using exterior point penalty functions. The hybridization of both techniques ASA and PSPSA provides a powerful hybrid heuristic optimization method; the proposed method is applicable to any problem where the topology of the structure is not fixed; it is simple and capable of handling problems subject to any number of nonlinear constraints. Extensive tests on the ASAPSPSA as a global optimization method are presented; its performance as a viable optimization method is demonstrated by applying it first to a series of benchmark functions with 2 - 50 dimensions and then it is used in structural design to demonstrate its applicability and efficiency.

Corresponding author. Tel.: 0096566610985; E-mail address: [email protected]

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A. El Hami, S. Kadry: Global optimization method for…

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1 Introduction The domain of design engineering is in constant progress and the race toward an optimal solution continues in its good train. The increasing need for optimum structural designs with the most efficient use of material without violating constraints has given rise to several developments in the past three decades. A good deal of effort has been centered toward the search of optimal structural designs [14]. In many of the previous studies, the authors have used local search algorithms. Such algorithms can only be successful if they are used to improve the current design if only a small segment of the boundary is allowed to move or if the objective function is convex. In fact, such assumptions do not hold, a design engineer working in the field of research and development has often to design completely new structures. The loading and support conditions of a particular design problem are usually known in advance, but the designer is not sure about what the final or optimal structure should look like. The essential goal of a designer in using an optimization algorithm is just to state the boundary conditions and let the algorithm perform some iterations without human intervention until it has automatically produced the best design. In this respect, the previous studies had only a relative success. Many of them relied too much on designer's intuition including the choice of initial design, or tight restrictions imposed on the movements of boundary. For these reasons, global optimization should take part in structural problems. In some of the previous studies, global search algorithms were used, but the accuracy remained questionable. Structural engineering and mathematical programming techniques should, both of them, collaborate in some way to develop a powerful and sophisticated programming system for structural optimization. This includes a robust optimization method coupled with modern tools of computer-aided design. This paper proposes a new hybrid simulated annealing algorithm for global optimization with constraints. Simulated Annealing (SA) has been developed rapidly since the past 20 years [5] as an effective and simple optimization technique, its superiority being in its good robustness and convenience to realize global optimization, whose conventional optimization techniques can seldom be attained. This algorithm, among few other

heuristics, is suitable for complicated problems where global optimum is hidden among many local optima. The idea of the method is an analogy with the way the molten metal is cooled and annealed. For a slowly cooled process, this system is able to find the minimum energy state. Such slow cooling is essential for achieving a low energy state. Although simulated annealing is a global optimization method, it suffers from some disadvantages such as inaccuracy or slow convergence to the global minimum. In order to overcome these concerns, many modifications have been proposed, viz.: (a) a new stopping rule and (b) the application of a local search method. The present article focuses on the enhancement of simulated annealing algorithms by proposing the new hybrid method ASAPSPSA. The method was designed in order to find the absolute minimum of an objective function without being sensitive to the starting point, itis capable of handling problems subject to any number of design variables or equality/inequality constraints. The present method can help researchers and practitioners devise optimal solutions to countless real-world problems. Numerical results demonstrate the efficiency, accuracy and applicability of the suggested method for structural optimization.

2 The basic simulated annealing The basic simulated annealing algorithm used in this paper is a simulated annealing (Fig. 1) based on the work of Van Laarhoven and Aarts [6], the algorithm is also well described by Sitarz [7]. A brief summary of the algorithm is given below for completeness. The following is a description of the periods of the algorithm in detail: 

Initialization: Generate initial solution s and initial parameters.



Termination: The algorithm terminates when T reaches the value of the minimal temperature allowed.


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

RAND (N(s)): Randomly choose a solution from N(s) where N(s) is the neighbourhood of s.



P (T; s; snew): P (T; s; snew) is the acceptance probability defined as: P (T; s; snew) = exp( E / k b .T ) where k b is the Boltzmann constant. E = E(snew) - E(s), is the amount of increase in the objective value caused by the uphill move and T is a parameter referred to as "annealing temperature".



Update T: Updating Temperature means cooling schedule. Tk temperature in the iteration k fulfils the following conditions:

Tk  0 and lim Tk  0 k 

The cooling schedule may be classic [8, 11] [9] Tk  T0  k , Logarithmic

Tk  T0 / log(k  1) and so on. Where  is some constant such that 0    1 , usually in the range of 0.90 - 0.99. T0 = 0.01, 1, 100.

probability P (T; s; snew). To avoid accepting large uphill move in the later stage of the search, the parameter T will be decreased over time by a schedule which is called "the cooling schedule". By accepting worse solutions with a certain probability, SA can avoid being trapped in a local optimum. There are two drawbacks for applying BSA Algorithm directly: one is its slow convergence; the other is its accuracy. To obtain faster convergence and to ensure that the region containing the optimal solution was found, we first accelerate and follow the new stopping rule and in the final stage, we apply PSPSA in the solution found by ASA to improve the final result. In this paper, a new ASA which is combined with PSPSA (local search) is proposed. From numerical experiments, ASAPSPSA shows its powerful ability of global optima searching and wide applicability.

3 The new proposed approach Although the BSA method approaches the neighborhood of the global minimum, it has, however, a difficulty in obtaining some required accuracy; it seems judicious to modify the basic algorithm according to a new stopping rule, and then to finish the algorithm with a faster convergent method. According to this idea, we modify BSA method to obtain the ASAPSPSA method as follows:

Figure 1. The basic steps of simulated annealing. Figure 2. The basic steps of ASAPSPSA. BSA Algorithm starts from an initial solution s at a high temperature T, and makes a series of moves according to RAND (N(s)). The change in the objective function values  E is computed at each move. If the new solution results in decreased objective function value, it is accepted with probability 1. If the new solution yields increased objective function value, it is accepted with

Commonly, structural problems are constrained. For the reason that SPSA method involves penalizing constraints, a penalty method is used; the constrained problem is then converted into an unconstrained problem which allows us to design a new method for constrained optimization problems, called Penalty Simultaneous Perturbation Stochastic Approximation (PSPSA) method.

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3.1 Penalty method Penalty method is a procedure for approximating constrained optimization problems by the unconstrained ones. The approximation is accomplished in the case of penalty methods by adding to the objective function a term that prescribes a high cost for violation of the constraints. Consider the problem

Minimize f ( x) , x  IR n  Subject to : x  S 

(1)

For large  , it is evident that the minimum point of problem (2) will be in a region where P is small. Thus, by increasing  , it is expected that the corresponding solution points will approach the feasible region S and, as subject to being close, will minimize f. Ideally then, as    , the solution point of the penalty problem will converge to a solution of the unconstrained problem. More generally, providing that the subset S is defined as

i  1,2,........., m   x : g i ( x)  0, S   and h j ( x)  0, j  1,2,........., p 

(5)

Where f is a continuous function in IR n and S is

then the problem (1) is equivalent to that of (2) with a penalty function of the form:

a constraint set in IR n . The idea of a penalty method is to replace problem (1) by an unconstrained problem of the form

P( x) 

Minimize f ( x)   P ( x)

(2)

where  is a positive constant and P is a function in IR n satisfying: (i) P is continuous (ii) P ( x)  0 for all x  IR n (iii) P ( x)  0 if and only if x  S

Supposing that S  x : g ( x)  0, the problem (1) can be replaced by the unconstrained one as follows:

Minimize f ( x)   g 2 ( x)    is large enough  x  IR n 

m

 (max [0, g ( x)]) i 1

i

2



p

 j 1

(3)

i 1,2,............., p

p

 (max [0, g ( x)]) i 1

i

2

(6)

SPSA is based on a highly efficient and easily implemented simultaneous perturbation approximation to the gradient: this gradient approximation uses only two loss-function measurements, independent of the number of parameters being optimized. The following general algorithm of SPSA is based on the work of J. Spall [10-11].

A very useful penalty function in this case is

P( x) 

2

3.2 SPSA method

Supposing now that S is defined by a number of inequality constraints

S  x : g i ( x)  0,

h j ( x)

(4) Figure 3. The basic steps of SPSA.


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4 Numerical results

4.2 Optimal design of a triangular plate

4.1 Benchmark test functions

The problem considered is a hexagonal steel plate, using thickness T1 and fillet radius FIL as the optimization parameters (see Figure 4). This problem uses a 2-D model and takes advantage of symmetry. The loading is tensile pressure (traction) of 50 MPa at the three flat faces. The purpose of this optimization problem is to minimize the volume V of the triangular plate without exceeding the allowable stress  von . The maximum stress anywhere in the triangular plate should not exceed 150 MPa. The optimum set of design variables can be achieved by solving the following minimization problem:

Several tests have been performed on some wellknown benchmark test functions; see Appendix, with known global optima in order to demonstrate the efficiency as well as the accuracy of the proposed method. Basic information about the benchmark functions are reported in Table 1, global optimization methods used for performance analysis are provided by Table 2 and computational results are summarized in Table 3 for each problem. Table 1. Basic information about 5 Benchmark functions

Minimize Volume  Subject to :  von  150  20  T1  40   5  FIL  15

Table 2. Global optimization methods used for performance analysis

(7)

In order to solve problem 1, we have applied two schemes: Scheme 1: both Structural Analysis and Optimization were done by using commercial software FEA code ANSYS (using First-Order method). Scheme 2: Structural Analysis was performed in commercial software FEA code ANSYS but an external Optimization code is used (ASAPSPSA optimizer).

Table 3. Performance of ASASPSA on 5 test functions.

4.3 Numerical results Calculations show that the results provided by ASAPSPSA are better than those provided by Scheme1 using First-Order method (see Table 4). Table 4. Scheme 1 vs. Scheme 2

As shown in Table 3, the ASAPSPSA can converge fairly close to the global optimum (verified by the analytical solutions). We also notice that, when compared with other methods, ASAPSPSA demonstrate its accuracy in achieving the global solution.

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4.4 Graphical results

5 Conclusion In this paper a new global hybrid method ASAPSPSA has been proposed. The new algorithm can be widely applied to a class of global optimization problems for continuous functions with box constraints. The experimental results show that the present method has proved the robustness and high performance of its algorithm. We can see clearly from numerical and graphical results that the algorithm can yield the global optimum with high accuracy.

Acknowledgements Figure 4. Triplate.

This work was supported by CMIFM, A.I. EGIDE, Number: MA/07/173. Appendix

RAn  10 n 

 x n

i 1

2 i

 10 cos ( 2  x i )



(8)

SHU   5   i cos  i 1

 i  1 x1  i    i cos  i  1 x2  i  5

  i 1

(9)



CA  Figure 5. The Graph of a Hexagonal plate with initial solution.

4 x12  2.1 x14 

(10) 1 6 x1  x1 x 2  4 x 22  4 x 24 3 n

DJ   xi2

(11)

i 1

Shekel = m

 i 1

1 , x  IR n (12) ( x  a i ) ( x  a i )  ci T

The Shekel function coefficients aij , ci are (Table 5):

Figure 6. The Graph of the optimal hexagonal plate using ASAPSPSA.


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Table 5: Shekel function-four-dimensional

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